Exponential variational integrators

ABSTRACT:We concentrate on physical problems where the kinetic and potential energy of their Lagrangian depend only on the generalized velocity and generalized position, respectively. Such systems are treated within the discrete variational integrators formalism.Towards this purpose we derive higher order exponential variational integration methods for the required numerical integrations. More specifically, for the derivation of such integrators, the action integral along the curve segment is defined by using a discrete Lagrangian that depends only on the end points of the interval. Then high order integrators are obtained by defining the discrete Lagrangian in any time-segment (on regular or non regular time grid) as a weighted sum of the Lagrangian at intermediate points.We also discuss PDEs that appear in various physical problems and can be written as multisymplectic systems, in which each independent variable has a distinct symplectic structure. For the derivation of the latter integrators, the necessary discrete Lagrangian is now expressed at the appropriate discrete jet bundle using triangle and square discretization.